In this paper dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community are considered. The cluster synchronization of these networks with or without time delay is studied by using some feedback control schemes. Several sufficient conditions for achieving cluster synchronization are obtained analytically and are further verified numerically by some examples with chaotic or nonchaotic nodes. In addition, an essential relation between synchronization dynamics and local dynamics is found by detailed analysis of dynamical networks without delay through the stage detection of cluster synchronization.

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Many realistic epidemic networks display statistically synchronous behavior which we will refer to as epidemic synchronization. However, to the best of our knowledge, there has been no theoretical study of epidemic synchronization. In fact, in many cases, synchronization and epidemic behavior can arise simultaneously and interplay adaptively. In this paper, we first construct mathematical models of epidemic synchronization, based on traditional dynamical models on complex networks, by applying the adaptive mechanisms observed in real networks. Then, we study the relationship between the epidemic rate and synchronization stability of these models and, in particular, obtain the conditions of local and global stability for epidemic synchronization. Finally, we perform numerical analysis to verify our theoretical results. This work is the first to draw a theoretical bridge between epidemic transmission and synchronization dynamics and will be beneficial to the study of control and the analysis of the epidemics on complex networks. Exploring the correlation between different dynamical behaviors which may appear in complex networks is an important and interesting pursuit on the way toward understanding them better. Among these different dynamical behaviors, synchronization and epidemic spreading on networks are closely related. For example, with the spread of an infective disease, people may reduce the frequency of inter-personal contact and take collective protective measures with increased regularity (for example, by washing hands frequently with clear water and soap, avoiding going to crowded place, and so on). This means that the spread of information between people during transmission of an epidemic disease can induce spontaneously collective risk-minimisation behaviour in spite of (or at least independently of) the actual disease pathology. Similar phenomena can also be found in many animal-borne infections. Hence, synchronization of individual's behavior and their epidemic behavior on networks can occur simultaneously and interplay adaptively. However, very little theoretical work has been done to consider these two dynamical behaviors together. In this work, we address this deficit. We will investigate mathematically the correlation between the dynamical synchronization and the epidemic behavior on complex networks for us to understand them in depth.